A Retrospective View of Hicks` Capital and Time

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A Retrospective View of Hicks’ Capital and Time: A NeoAustrian Theory
by
Edwin Burmeisteri
1. Introduction
In 1973 Sir John Hicks published Capital and Time: A Neo-Austrian Theory. This
was his third book with the word “capital” in its title, the first being his classic Value and
Capital [1939] and the second being Capital and Growth [1965]. It departed significantly
from his earlier work by assuming that the technology of an economy consisted of a set
of neo-Austrian production processes in which a time sequence of inputs at produces a


time sequence of outputs bt .
In June, 1974 I published a review article in the Journal of Economic Literature
entitled “Synthesizing the Neo-Austrian and Alternative Approaches to Capital Theory:
A Survey” using Hicks’ book as a filter to select a list of topics for discussion.ii Now,
with almost 30 years of hindsight, I will revisit some of the problems that, in my view,
remain both unsolved and important.
First, however, I point out that reading Hicks in 2007 is as much a delight as it
was in 1973. His style is refreshingly old-fashioned, focused on answering economic
questions with mathematics used only as a means of achieving those answers. For
example, Hicks’ view of the von Neumann model is that it is logically elegant, but that
“… the categories with which it works are not very recognizable as economic categories;
so to make economic sense of its propositions translation is required. One has got so far
away from the regular economic concepts that the translation is not at all an easy matter”
[Hicks, 1973, p. 6].
But, as I shall argue below, this bias in favor of “regular economic concepts”
comes at the cost of generality. As a result the primary contributions of Capital and
Time: A Neo-Austrian Theory are pedagogical. Hicks’ elegant examination of simple
problems serves to deepen our economic understanding of many capital theory principles
such as, for example, the duality between the factor-price frontier and the optimal
1
transformation frontier. Yet one is left with a sense of unease that treacherous territory
may lie just beyond these simple examples.
2. The Hicks Neo-Austrian Technology and Truncation
The Hicks technology is based on the Austrian tradition of Böhm-Bawerk,
Wicksell, and Hayek in which a flow of inputs over time produces output at a later point
in time. Hicks generalizes this idea so that a production process consists of a time
sequence of inputs at that produces an associated time sequence of outputs bt . The


Hicks neo-Austrian technology is the set of all such feasible production processes.
It is assumed that homogeneous labor is the only input and that there is only one
type of homogeneous output, which Hicks identifies simply as “goods,” although I prefer
to interpret output as the quantity of a single type of consumption good [Hicks, 1973, p.
37]. Using this single consumption good as the numeraire, bt measures both the physical
quantity and the value of output. Then letting wt denote the real wage rate (in terms of
the consumption good) during period t, a production process yields a net output stream
given by
(2.1)
q 
n
t
t 0

b  w a 
n
t
t t
t 0
.
More than one production process may be used during any time period, and over time the
economy may or may not converge to a steady-state equilibrium in which one most
profitable production process is employed.
Hicks also assumes that at  0 and bt  0 for t  0,1,K ,m  1 . He then defines
the construction period as the m time periods t  0,1,K ,m  1 during which labor inputs
are employed, but there is no output of the consumption good [Hicks, 1973, p. 15]. It
suffices here to point out the obvious fact that this technology is extraordinarily simple;
see Burmeister [1974] for some details. However, it does enable Hicks to focus on some
of the economic questions that arise from the pure role of time, without, for example,
having to deal with the complications of heterogeneous inputs and outputs.
There is often some ambiguity in discrete-time models because the end of one
time period coincides with the beginning of the next. To avoid inconsistencies in the
Hicks neo-Austrian model, one can interpret labor input for a production process as the
number of workers employed at the beginning of period t , who then work for the whole
period and are paid a real wage rate at the end of period t . The output of the consumption
good, on the other hand, is realized only at the end of period t . The reasons for these
timing conventions are explained elsewhere [Burmeister, 1974, pp. 417-418] and need
not concern us here.
2
Now assume that the real-wage rate is constant, wt  w , and also that there is a
constant per period (Hicks uses weeks) real rate of interest (in terms of the consumption
good) denoted by r . Then the capital value of the process at the beginning of period 0 as
n
k0   qt R
(2.2)
(t 1)
t 0
n


  bt  wat R(t 1)
t 0
where the interest rate factor is R  1 r . Note that k0 is simply the present discounted
value of the production process. Hicks assumes that capital markets are in equilibrium so
that k0  0 [Hicks, 1973, p. 32].iii Note that k0  0 is equivalent to a zero-profit condition
when all inputs and outputs are measured in terms of discounted prices.
More generally, the capital value of the process at the beginning of any time
period t is
n
kt   qi R
(2.3)
(i1t )
n


 bi  wai R(i1t )
it
it
so that


kt  qt  kt 1 R 1 .
(2.4)
Thus the capital value of the production process at the beginning of period t is equal to
the value of net output at the end of period t , denoted by qt , plus the capital value at the
beginning of period t  1 , both discounted.iv
Given the real wage rate w and the interest rate r , the economic lifetime of a
project,  , is determined by maximizing (2.2) over the terminal time n under the very
strong assumption that the production process can by truncated at any time. That is, a
feasible production process
a ,b 
n
t
t
t 0
can be truncated if, and only if, the shorter
process
(2.5)
a, b  a ,b ,a ,b ,K ,a ,b ,0,0,K ,0,0
n
t
t
t 0
0
0
1
1
m
m
is also technologically feasible for all m  n . It is important to note that the conventional
free disposal assumption does not imply the truncation property (2.5). Free disposal
implies that the same or less output can be produced with additional inputs, but it implies
nothing about what can be produced with fewer inputs.
3
Denote the present discounted value of the project at the beginning of period 0,
when it is operated through to period T, as
(2.6)


T
T
t 0
t 0


k 0,r, T ; w   qt R(t 1)   bt  wat R(t 1)
for T  0,1, , n .
Hicks denotes this value by the ambiguous notation k0 , as we shall do when no confusion
is possible.v We assume that the process is viable at the given real wage rate and interest
rate so that this present discounted value is strictly positive for some value of T.
The optimal lifetime or duration of a production process is defined as a value of
 for which
(2.7)

 
k 0,r, ; w  k 0,r, T ; w

for all T   .
Using the Hicks notation, this definition rules out ties and implies that
(2.8)
kt  0 for all t  0,1,K , ;
see [Hicks, 1973, p. 18 and especially footnote 2].
3. Hicks’ Fundamental Theorem
Given all of the stated assumptions, Hicks proves what he calls a Fundamental
Theorem, namely given a real wage rate, a rise (fall) in the rate of interest will lower
(raise) the capital value of the production process for all time horizons 0  t   [Hicks,
1973, p. 19]. The following numerical example proves that this result does not hold
without the truncation property. Consider a production process with net outputs
q0  1, q1  2.3, q2  1.32 at some given real wage rate. The capital values





kt r computed from (1.3) for interest rates r  0, 0.1, 0.15, 0.2, 1 are given in Table 1.
4
INSERT TABLE 1 HERE
Note that as the rate of interest is increased, the capital value at the beginning of period 1
does not always fall, but rather it first rises and then falls. Because q2  0 , it is more
profitable to truncate the process at the beginning of period 2 no matter what the interest
rate. This fact is also reflected by the negative values of k2 r . Note also that both

k0 (0.1)  0 and k0 (0.2)  0 so that this untruncated process has two internal rates of
return. It will be operated only for interest rates satisfying 0.1  r  0.2 .
When truncation is allowed, the optimal lifetime is   1 . Table 2 shows the
corresponding capital values. Note that now both the capital value at the beginning of
period 0 and the capital value at the beginning of period 1 fall as the interest rate
increases, as asserted by Hicks’ Fundamental Theorem.
INSERT TABLE 2 HERE
It follows immediately from the Fundamental Theorem that if there exists a
positive rate of interest r  r1 for which k0  0 , then this value r1 is unique. In addition,
such an r1 always exists if the production process is viable at the prevailing real wage
rate, and it is equal to the internal rate of return for the production process [Hicks, 1973,
pp. 18-21]. Although Hicks’ exposition of his Fundamental Theorem is both elegant and
based on economic principles, the result had been anticipated by others who derived
sufficient conditions for the uniqueness of the internal rate of return in much more
general cases than the one considered by Hicks; see, for example, [Arrow and Levhari,
1969], [Flemming and Wright, 1971], and especially [Sen, 1975].
This Fundamental Theorem is illustrated in Figure 1. For this numerical example

the production process has net outputs qt
4
t 0


 1, 0.6, 0.75,  0.9, 1 . At interest rates
r  0 and r  0.1 the optimal lifetime of the process is   4 , while at interest rates
r  0.15 and r  0.216515 it is   2 . As the interest rises from 0, the capital value
curves fall. Therefore there exists exactly one interest rate, r  0.216515 , for which
k0  0. This interest rate is the unique internal rate of return for the optimally truncated
process.
INSERT FIGURE 1 HERE
For completeness in Appendix A we sketch a proof of Hicks’ Fundamental
Theorem using our timing convention for inputs and outputs.
5
4. The “Cambridge Capital Theory Controversies”
No historical account of capital theory is complete without mention of the socalled “Cambridge Capital Theory Controversies” that peaked in the mid 1960’s. We
need to first briefly remind ourselves of a few key features of this controversy before we
can see how Hicks’ Capital and Time fits into the picture.
The issue concerns non-joint production, constant-returns-to-scale technologies
with one primary factor, labor. Many people once believed that many of the steady state
results obtained for a one-capital good Solow/Swan model also held for more complex
technologies with many different types of capital goods. In particular, by analogy with
the familiar one-capital good models, some thought—incorrectly as it turns out—that
steady-state per capita consumption always increases with decreases in the steady-state
rate of interest, so long as the rate of interest remains above the golden rule value r  g
(where g is the exogenous growth rate of labor). The mistaken intuition for such
thoughts was based on the notion of “capital deepening” whereby an economy in a
steady state with a low rate of interest (but still bigger than g ) would have more “capital”
than at a higher rate of interest, and hence it would be able to produce more per capita
consumption. The latter statement is, of course, correct for models with only one type of
capital good.
Numerical examples originating from the Cambridge, England, school of thought
immediately convinced those in Cambridge, Mass. that such “capital deepening” results
do not necessarily hold in a world with more than one capital good. Figure 2 shows the
factor-price frontiers for production processes A and B. These frontiers trace the steadystate relationship between the rate of interest and the real wage rate. For any given
interest rate, competition insures that the corresponding equilibrium wage rate is on the
highest frontier using the corresponding production process. Alternatively, as Hicks
prefers, one can take w as given, and then the corresponding point on the factor-price
frontier gives the equilibrium value of r .
The processes A and B have switch points at interest rates r1 and r2 . Therefore
process A is used for all interest rates satisfying either 0  r  r1 or r2  r  rmax , where
rmax is the largest steady-state interest rate for which process A is viable. Similarly,
process B is used for all interest rates satisfying r1  r  r2 , and at the two switch points,
both process can coexist. Thus Figure 2 illustrates what is called the reswitching of
techniques, though here I use the term “process” instead of “technique” to be consistent
with Hicks.
6
INSERT FIGURE 2 HERE
Denoting the factor-price frontiers illustrated in Figure 2 by w  f A (r)
and w  f B (r) , and assuming that labor does not grow, it is easily shown that per capita
consumption is equal to c  f A (0) when process A is employed and is equal to c  f B (0)
when process B is employed. Therefore, the so-called paradoxical steady-state
consumption behavior illustrated in Figure 3 exists. That is, as the steady-state interest
rate decreases from, say, 0.8 to 0.2, the economy switches from using production process
A to production process B and steady-state consumption falls. This clearly demonstrates
that no notion of “capital deepening” can be valid in models with more than one capital
good.
INSERT FIGURE 3 HERE
Moreover—and there has been much confusion about this point—while the
existence of reswitching clearly reveals the existence of such paradoxical consumption
behavior, this behavior can arise even in models for which there is no reswitching. This is
illustrated in Figure 4. Here process C is employed at high interest rates, process B is
employed at intermediate interest rates, and process A is employed at low interest rates.
Yet steady-state consumption decreases as the interest rate is lowered from, say, 0.8 to
0.2.
INSERT FIGURE 4 1 HERE
In retrospect it is clear that steady-state comparisons are not of very much
economic interest because they do not represent choices among feasible alternatives.
Given that an economy is in a steady-state equilibrium at some interest rate, the relevant
economic questions concern the properties of the feasible dynamic paths starting from
this steady state as an initial condition. And there is nothing strange or paradoxical about
these feasible paths. Neoclassical economics is alive and well, even when there are many
different types of capital goods.
7
5. The Factor-Price Frontier and Duality for the Hicks NeoAustrian Model
We now take as given any real wage rate wi for which a production process is
viable, and let ri denote the corresponding unique internal rate of return for which the
capital value at the beginning of period 0 is equal to 0, the condition for equilibrium in
Hicks’ capital market. The set of such equilibrium points wi , ri defines a function



r  w
(5.1)
which Hicks calls the efficiency curve of the process. He objects to the more common
terminology factor-price curve on the grounds that the interest rate is not the price of a
factor.vi Nevertheless, there is considerable advantage to using terminology consistent
with the main body of economic literature, and here we shall refer to the relationship
defined by (5.1) as the factor-price curve.


More precisely, since it is easily shown that   w  0 ; hence  w may be
inverted to write our factor-price curve in the conventional form
(5.2)

w f r
with

f  r  0 .vii
Such a factor-price curve exists for every production process. The outer envelope of the
set of these factor-price curves defines the factor-price frontier for the economy, which is
denoted by
(5.3)

w F r
with

F r  0 .
For any given feasible interest rate r, an economy in competitive equilibrium always acts
to maximize the real wage rate w by operating on this factor-price frontier.
Along a factor-price curve and the factor-price frontier, duration is always at its
optimal length. Hicks also defines a restricted efficiency curve along which the optimal
duration is fixed at the value appropriate for some interest rate [Hicks, 1973, pp. 66-67].
Our corresponding restricted factor-price curve is denoted by
(5.4)
w   (r) .
For a given production process, the optimal duration may change with the interest rate.
For each optimal duration, there is a different w   (r) function, and their outer
envelope is the factor-price curve given by (5.2).
8
In addition, using the capital market equilibrium condition k0  0 we may
compute (5.4) as
1
(5.5)
w   (r) 
 b 1  r 
t 0
1
1
(t 1)
t
 a 1  r 

(t 1)
t
t 0
 b 1  r 
t 0
1
t
t
 a 1  r 
t 0
t
t
where 1 is the optimal duration at the interest rate r1 .
We now need to examine the quantity side of the model before we can discuss the
duality results. Hicks assumes that there are constant-returns-to-scale in the sense that
production processes can be replicated. Let  denote the beginning of the current time
period (“week”), and let x t denote the number of unit production processes that began

operation t periods ago. Let either w  w1 or r  r1 be given, where w1  f r1 , and let
1 be the corresponding optimal duration. The total labor input and the current output
(of the consumption good) are
1
A   x t at
(5.6)
t 0
and
1
B   x  t bt ,
(5.7)
t 0
respectively. In a steady-state equilibrium the number of starts must grow at the constant
rate of g per period and hence

x  x0 1  g
(5.8)


where x0 determines the scale of the system. Then (5.6) and (5.7) are replaced by
1
(5.9)

A   x0 1 g
t 0

 t
 1
at  x0   at 1 g
 t 0

and
9

  1 g 
t


1
(5.10)

B   x0 1 g
t 0

 t
 1
bt  x0   bt 1 g
 t 0


  1 g  ,
t


respectively. Similarly, in a steady-state equilibrium the current value of capital is
(5.11)
1
1
t 0
t 0

K   x t kt   x0 1 g

 t
 1
kt  x0   kt 1 g
 t 0


  1 g  .
t


From (5.9) and (5.10) we see that per capita consumption is
1
(5.12)
c
C B


L
A
 b 1  g 
t
t 0
1
t
 a 1  g 
t 0

 g .
t
t
This is Hicks’ duality result: Constant per capita consumption depends on g in exactly
the same way that the real wage rate depends on r in (5.5). The Golden Rule theorem that
c  w at r  g and other well-known results follow trivially.
The duality between steady-state per capita consumption and the real wage rate is
fundamental, and the Hicks neo-Austrian model is only one example of it. The result was
first proved by von Weizsäcker [1963] and subsequently was developed by Bruno [1969]
and Hicks [1965] for Leontief technologies. Burmeister and Kuga [1970] proved the
result for a neoclassical model with heterogeneous capital goods. The concept of a factorprice frontier originates with Samuelson [1957], and he provides an excellent review in
[Samuelson, 1983, pp. 464-471].
6. Roundaboutness and Other Doomed Austrian Concepts
Consider three alternative production processes:
14, 0,14, 0, 14, 0,14,00,10
Process B = 20, 0, 20, 0, 20, 00,10
Process C = 35, 0, 35, 00,10  .
Process A =
Then using (5.3) or (5.4) we compute that Process A is used for 0  r  0.137009 , Process
B is used for 0.137009  r  0.318729 , and Process C is used for r  0.318729 .
One of the many possible definitions of the “average period of production” or
“roundaboutness.” is
10


 t  1a
t
t 0

a
t 0
,
t
and for these three processes we have
 A  2.5
B  2
 C  1.5 .
This provides an example of:
Property 1:
The “degree of roundaboutness” increases as the interest rate falls.
Now taking the simple no-growth case with g  0 , we can compute from (5.12)
that the steady-state per capita consumption levels for the three processes are:
c A  0.178571
c B  0.166667
cC  0.142857 .
We thus have an example of:
Property 2:
Per capita consumption rises as the interest rate falls.
Finally, the current value of capital with g  0 is computed from (5.11):
K A  21.646980
K B  15.659341
K C  10.119048 .
This gives us:
Property 3:
Capital values rise as the interest rate falls.
These three Austrian properties hold only for special cases such as the one given
here and are not generally valid. By 1973 when Hicks wrote Capital and Time, he was
quite aware of this fact.viii Indeed, the essence of Hicks’ approach is to generalize the
Austrian notion of output at a point in time to one in which output is a stream over time,
and with this generalization the whole notion of “roundaboutness” collapses [Hicks,
1973, pp. 8-9].
11
In Chapter IV Hicks acknowledges that his production processes can exhibit
reswitching, exactly as we have described it in Section 4 above. The existence of
reswitching immediately provides a counterexample to Properties 1, 2, and 3 above. He
calls reswitching “a curiosum,” yet acknowledges that its importance is “… for the much
more substantial issue that lies behind it.” [Hicks, 1973, p. 41]. And that substantial issue
is that the rate of interest cannot be used to characterize differences among production
processes. Hicks does not seem to recognize that the failure of Properties 1, 2, and 3 can
arise even in models for which no reswitching exists, as illustrated by our Figure 4;
Bruno, Burmeister, and Sheshinski [1966] first made this observation and explained why.
The Hicks solution is to assume that every production process has a Simple Profile,
thereby circumventing all of these difficulties [Hicks, 1973, pp. 41-42].ix
Some bad ideas have amazing endurance, and “roundaboutness” seems to be one
of them. For example, Yeager’s 1976 paper “Towards Understanding Some Paradoxes in
Capital Theory” won a prize for the best Economic Inquiry publication in that year. Yet it
is wrong, as Yeager graciously and candidly acknowledged in 1978; see [Yeager, 1976]
and [Burmeister and Yeager, 1978].x
Earlier, before the possibility of reswitching was recognized, Samuelson fell into
the similar error of asserting that economies with a lower steady-state interest rate had
more “capital” and therefore were able to produce more per capita consumption, though
with diminishing returns; see [Samuelson, 1964, Appendix to Chapter 28: Interest and
Capital, pp. 594-600]. Samuelson has also candidly admitted his mistake; see especially
[Samuelson, 1966].
Two more obscure Austrian-like properties deserve mention:
Property 4:
The economic lifetime of a machine increases with decreases in the interest rate.
Property 5:
The optimal durations for production processes in use increases with decreases in
the interest rate.
Properties 4 and 5 also are not generally true. Hagemann and Kurz have constructed a
clever example—but more complex than the ones we have considered here—that shows
them both to be false; see [Hagemann and Kurz, 1976] and [Kurz and Salvadori, 1995,
pp. 212-216].
12
7. The Neo-Austrian Approach as a Generalized von
Neumann Model
I have summarized a generalized von Neumann model in Appendix B. Now I
provide a simple example to illustrate how the Hicks neo-Austrian model is but a special
case of this more general framework.
  
  
Consider the production process a1 , 0 , a2 , b2 , a3 , 1 . The generalized von
Neumann interpretation is that Activity 1 uses a1 workers employed for one period
produce one new machine; Activity 2 uses a2 workers together with one new machine
employed for one period to produce jointly one one-year-old machine and b2 units of the
consumption good; and Activity 3 uses a3 workers together with one one-year-old
machine to produce one unit of the consumption good. In general, Activity j  1 is
associated with the jth period of production in Hicks’ notation. Therefore the optimal
duration problem is automatically solved once we identify which activities are operated at
positive intensity levels in the von Neumann solution.
In the notation of Appendix B, we have


(7.1)
A0  a1 a2 a3 ,
(7.2)
0 1 0 


A  0 0 1  ,
0 0 0 


and
(7.3)
1 0

B  0 1
0 b
2

0

0 .
1 
We assume that all three activities are operated at a positive intensity level for the
given wage rate w and interest rate r. Then equation (B.10) in Appendix B implies
(7.4)
 
wA0  1  r A  pB
or
(7.5)
 
wA0  p  B  1 r A .
13
This example is especially easy to solve because the number of activities is equal to the
 
1
number of goods. Therefore for given r such that  B  1 r A exists, (7.5) can be
solved for

 
1
p r  wA0  B  1 r A .
(7.6)
The equation for the factor-price frontier in terms of the consumption good is
W (r) 
(7.7)
w
.
p3 r

For an economy operating in equilibrium on this factor-price frontier, equilibrium prices
in terms of the consumption good are


 
P e r  W r A0  B  1 r A
(7.8)
1
and the price of the consumption good is P 3e(r)  1 as in Hicks.
Using (7.6)-(7.8) to compute P 3e(r) , we find that P 3e(r)  1 is equivalent to the
Hicks condition for equilibrium in the capital markets:
(7.9)

0  W r a1
1 r


1 r 
b2  W r a2
2


1 r 
1 W r a3
3
 0  k0 .
Similarly, we can show that the value of the production process at the beginning
of period 2 is
(7.10)


P e2 r  k  2, r, 2; W r 
where the right-hand-side of (7.10) is defined by equation (A.1) in Appendix A.xi Written
out in full equation (7.10) may be expressed as
(7.11)
   
W r a1 1 r  W r a2  b2 

1 W r a3
1 r
.
The left-hand-side is the net cost of producing a one-year-old machine. The right-handside is the present discounted value of the future revenue it can produce, which is also the
value of the production process at the beginning of period 2. In equilibrium these two
values must be equal.
14

If P e2 r  0 , as it will be if a3 
1

, then Activity 3 is shut down and the Hicks
W r
production process is truncated. This example also provides another counterexample to
Austrian Property 5 stated in the previous Section 6. Suppose a3  1 and W 0  1 so that

Activity 3 is not profitable at low interest rate. Since
   0 , there is some interest
dW r
dr
rate r1 for which W r1  1 . Activity 3 becomes profitable at high interest rates

with 1 
1
for r  r1 . Thus the optimal length of the production process increases with
W r
an increase in the interest rate.xii

We conclude that the Hicks neo-Austrian approach is but a special case of the von
Neumann approach. Additional details and discussion are contained in [Burmeister,
1974]. Kurz and Salvadori [1995] provide superb analyses of more complex
technologies, including, for example, models with fixed capital, joint production, jointly
utilized machines, and land.
8. The Necessary and Sufficent Condition for a “WellBehaved” Economy Across Steady-State Equilibria
Once it is recognized that nothing of economic substance is to be gained from the
neo-Austrian approach to capital theory, we turn to a more general question:
Under what circumstances will an economy be “well-behaved” in the
sense that steady-state per capita consumption always rises (falls)
with decreases in the rate of interest, provided r  g ( r  g )?
The answer to this question has a surprisingly simple characterization. Provided joint
production of final consumption goods is excluded, it has been shown that a neoclassical
economy with a single consumption good and N different types of heterogeneous capital
goods is “well-behaved” in the above sense if, and only if,
 p r 
i
i1
  0
dki r
N
(8.1)
dr
15
for all feasible r ,

where ki r denotes the steady-state per capita quantity of the ith capital good at interest
rate r. This result was first proved by Burmeister and Dobell [Burmeister and Dobell,
1970, Theorem 7, pp. 286-287]; see also [Burmeister and Turnovsky, 1972].

 
N
Denoting the value of capital by v r   pi r ki r , we have:
i1
(8.2)
Total Wicksell Effect 
N
(8.3)
Price Wicksell Effect  
i1
 ,
dv r
dr
 k r ,
d pi r
dr
i
and
N
(8.4)
Real Wicksell Effect   pi (r)
i1
 .
dki r
dr
If prices are in terms of the wage rate, the Price Wicksell effect is always positive. And it
can be so positive that it outweighs a negative Real Wicksell effect. But this does not
matter. No matter how the per capita value of capital behaves, it is the sign of the Real
Wicksell Effect that is of fundamental economic significance. Another way to see this
fact is to consider new prices measured in terms of the consumption good. The sign of
the Real Wicksell Effect is not influenced by this change (or any other change), although
the sign of the Price Wicksell Effect and hence of the Total Wicksell Effect may be
altered.
This result generalizes to non-differentiable technologies such as our generalized
von Neumann model. Suppose that Technique A is viable for g  r  r S , Technique B is
viable for r  r S  g , and both are viable at the switching point r  r S with p A  p B  p ,
with p is a row vector in terms of the single consumption good. Define the change at the
switch point in per capita consumption c  c A  c B , and define the corresponding change
in per capita capital stocks by the column vector k  k A  k B . It can then be shown that
per capita consumption rises as the rate of interest falls from above r S to below r S and
there is a switch from Technique A to Technique B if, and only if,
(8.5)
p k  0 ,
which is a generalization of (8.1) to non-differentiable technologies. Proofs are contained
in [Burmeister, 1976] and Burmeister [1974, p. 453].
16
We see, therefore, that the Hicks Simple Profile assumption plays the role of a
sufficient condition ensuring that the Real Wicksell Effect in his model (if the
intermediate capital goods are properly defined as in our von Neumann generalization) is
always negative.
But, historical concerns aside, why should we even care about whether or not an
economy is “well behaved” across alternative steady-state equilibria? After all, in Section
4 we argued that comparisons of steady-state equilibria are not of great economic interest
because they do not represent the viable alternatives open to an economy.
I can think of only one reason. Sometimes it may be possible to interpret
economic data as having been generated from an economy in alternative steady-state
equilibria. In such cases it would be comforting if a rigorous theoretical foundation could
be found for the existence of an aggregate production function of the type so often used,
or misused, in econometric work. In a long-term research project now stretching out over
more than ten years, I have established a near aggregation result: an index of aggregate
capital and a corresponding well-behaved per capita aggregate production function exist
across steady-state equilibra if, and only if, the Real Wicksell Effect is negative at all
feasible interest rates [Burmeister, 2008].xiii This result falls short of full aggregation
because this per capita aggregate production function does not correctly predict the real
wage rate.
Ironically, one of the few known sufficient conditions for negative Real Wicksell
Effects is a generalization of the Marx Equal Organic Composition of Capital condition,
in which case full aggregation exists; see [Burmeister, 2008].xiv
9. Dynamics and Technological Change
Normally one thinks of a growth path as starting from arbitrary initial conditions.
Instead, Hicks restricts his attention to paths that start in some steady-state equilibrium,
and he calls the resulting dynamic path a Traverse. The economy moves from this initial
steady state because there is some technological change that shifts the factor-price
frontier outward and consequently makes a new process more profitable at the previous
rate of interest. He considers two distinct cases: The Fixwage Path [Hicks, 1973, Chapter
VIII, pp. 89-99] and The Full Employment Path [Hicks, 1973, Chapter IX, pp. 100-109].
Along a Fixwage Path, the real wage rate is given, and hence the rate of interest is
determined from the factor-price frontier. Along a Full Employment Path, the interest rate
g
is determined by the condition r  where s is the fixed savings propensity out of profit
s
income, so there is no maximizing behavior on the part of consumers. In both cases Hicks
assumes that all production processes have the Simple Profile described in Footnote 7.
17
Now let asterisks denote the steady-state equilibrium before the technological
change. Hicks then defines the index measuring technological change
(9.1)

I r 
 ,
w r 
w r

which he calls “… an Index of Improvement in Efficiency, in one sense or another
[Hicks, 1973, p. 75, italics in the original].” Within this restrictive framework, Hicks is
able to revisit the question of Ricardo on machinery [Ricardo, 1911, Chapter XXXI].
Ricardo claimed that the introduction of machinery could have an adverse effect on the
total wage bill in the short run. Hicks is able to find an interpretation of his model for
which this conclusion is correct [Hicks, 1973, pp. 98-99].xv
However, as I pointed out in [Burmeister, 1974, pp. 436-437], the set of possible
outcomes increases when more than one primary factor of production exists. For
example, consider a Ricardo model with two primary factors of production, labor and
land, and let the real rental rate for land (in terms of the single consumption good) be
denoted by  . Then (again ruling out joint production and assuming constant returns to
scale) there exists a factor-price surface
(9.2)

r   w, 

defined across steady-state equilibria. Moreover, the trade-off between w and  for given
r is quasi-convex.xvi Accordingly, when “the introduction of machinery” shifts 
outward from the origin, it is possible for the economy to settle down in a new steady
state with a higher real wage rate, a lower interest rate, and a lower rental rate for land.
Thus, without more being said about the model, one cannot rule out the possibility that an
innovation helps workers at the expense of the owners of both capital and land.
The Hicks analysis of dynamic paths is further simplified by his assumption of
“… static expectations—that the wage that is ruling at [the current time] is expected to
remain unchanged, at least so long as the processes started at [the current time] are
expected to continue [Hicks, 1973, p. 110].”xvii Given these extraordinarily strong
assumptions about technology and behavior, and sometimes using yet additional
assumptions, Hicks is able to prove that some paths converge to a new steady-state
equilibrium. The analysis is tedious and full of details about the characteristics of the
dynamic paths. If these details represented general economic properties, some of them
might be of considerable economic interest. But they do not hold in general. They have
been shown to hold only for very special and economically unappealing cases.
What do we know in general about the economic properties of the sort of dynamic
paths considered by Hicks? Many results were established in the decade after Capital and
Time was published. Bliss [1975] is a good starting point, and Burmeister [1980, Chapter
5] provides an introductory exposition with references to some of the more technical
literature. One of the most important theorems to emerge is contained in two classic
18
papers by Cass [1975a and 1975b] where he proved the necessary and sufficient
conditions for consumption efficiency over an infinite time horizon.
Three results merit special mention because they concern the kind of economic
questions in which Hicks was most interested:
First, consider the intertemporal production possibility frontier for a constantreturns-to-scale competitive economy:
(9.3)


f c0 ,c1 ,K ,cT ;k0 , kT  0
where ct denotes per capita consumption and the beginning of period t, k0 is a vector of
the N heterogeneous capital goods (per capita) at the beginning of the initial period, and
kT is a vector of the terminal capital stocks. Both k0 and kT are given. Then the interest
rate over period t is given by
(9.4)
rt  
ct 1
ct
 1.
f 0
If f is not differentiable, rt is bounded by right- and left-hand partial derivatives.xviii This
result stems from the work of Irving Fisher and is developed in many of Samuelson’s
writings. In particular, an interesting account of the connection between this correct result
and the mistake that was revealed by the existence of reswitching, see [Samuelson, 1966,
footnotes 6, 7, 8, and 9].
Second, a result similar to the first holds for the real wage rate. With a nondifferentiable technology such as the generalized von Neumann model discussed here, at
every point in time the technology and competitive equilibrium impose upper and lower
bounds on the possible values of the real wage rate. Within these bounds—but only
within these bounds—there is room for a theory of income distribution that depends, for
example, on the power of labor unions. Some of these results are contained in
[Burmeister, 1984].
And third, the steady-state equilibrium for dynamic models with more than one
capital good is usually a saddlepoint. Therefore the models converge only if there is some
economic mechanism for determining the proper initial conditions to put the economy on
the stable manifold. Some of the original papers addressing this problem include
[Burmeister, Caton, Dobell, and Ross, 1973], [Hahn, 1966], [Malinvaud, 1953],
[Samuelson, 1967], [Shell and Stiglitz, 1968], [Burmeister and Graham, 1974], and
[Brock, 1972].
19
10. Lack of Impact and Unresolved Questions
Capital and Time has had little enduring impact on the economics profession, as
revealed by the citation count in Table 3 for the twenty-eight years from 1980 to 2007.
The reason, I believe, is not so much because of Hicks’ neo-Austrian approach, but rather
because the kind of analysis contained in Capital and Time became technically obsolete
soon after its publication in 1973. New tools were developed, and it changed the kind of
economic models that we build. Rational expectations should be used and the Hicks’
assumption of static expectations is no longer acceptable. The technology specification
should include stochastic shocks. Both producers and consumers should exhibit
maximizing behavior under uncertainty. Technological change should not be exogenous,
but rather should arise as the result of economic decisions. But these were not common
features of economic models in 1973.
It appears that Capital and Time represents the continuation of a research agenda
laid out by Hicks in the second edition of The Theory of Wages [Hicks, 1963].xix For
example, in the section “Wages, Interest, and Growth” [Hicks, 1963, pp. 363-372], he
explored the relationship between the real wage rate and the interest rate and asked how
an economy can move from one steady-state equilibrium to another. Of particular interest
was the question, “Under what conditions are labor unions able to raise the real wage
rate?”
In my opinion, therefore, Hicks probably had two primary objectives in writing
Capital and Time:
1. To clarify the pure role of time in economics, without the complications of
uncertainty, by providing an alternative to the standard production function that
would help us to better understand the determination of real wage rates.
2. To establish the dynamic stability properties for models using his alternative neoAustrian technology.
In pursuing these objectives, Hicks brought to bear the tools that were available to him at
the time. This resulted in a model containing many features, as noted above, that simply
are unacceptable by modern standards. Consequently while he made considerable
progress toward achieving these two objectives, he fell far short of what today we would
deem to be success.
But we should not be too harsh judging Hicks. The fundamental questions that
might have been partially answered had Hicks achieved his objectives remain essentially
unresolved today:
First, many fields of economics continue to use models with aggregate capital
and an aggregate production function even though these have no rigorous theoretical
foundation, except under extraordinarily restrictive assumptions. Other aspects of these
modern models are often very sophisticated—but aggregate capital is a shaky logical
foundation upon which to build. I do not mean to imply that these models are “wrong.”
20
All models are of necessity no more than a caricature of reality, and their usefulness
depends upon whether or not they are able to shed light on interesting economic
problems. By this criterion many models using aggregate capital are clearly a success.
We will continue to see such models until someone discovers a better alternative, and that
is how research should progress. But the Hicks neo-Austrian technology did not provide a
better alternative.
Second, the dynamic stability properties of models containing more than one
capital good have not been completely worked out. We know a lot about special cases,
especially when there is maximizing behavior and rational expectations. But in general it
is difficult to establish stability and to rule out cycles, even without the realistic
complications introduced by uncertainty.
I conclude that despite Hicks’ attempt to address questions of fundamental
economic importance, he was unable to shed much light on them. And so, I am afraid, for
the most part the economics profession has simply ignored Capital and Time. Yet it
remains a delightful book to read, and one can still learn a lot of economics from it.
INSERT TABLE 3 HERE
21
Appendix A
For completeness we sketch a proof of Hicks’ Fundamental Theorem using our
timing convention for inputs and outputs.
Step 1:
Given a real wage rate, the value of a production process at the beginning of
period t, when that process is operated through to period T, is given by
T
k(t, r, T ; w)   qi (1 r)(i1t ) , t  T  n .
(A.1)
it
Taking the interest rate r as given for now, it is assumed that there exists some duration T
for which the process is strictly viable:
k(0, r, T ; w)  0 .
(A.2)
Step 2:
If the process is operated for its optimal duration,  , the capital value at the
beginning of period 0 is maximized and by (A.2) this value is positive:
k(0, r, ; w)  0 .
(A.3)
Also by definition of 

 
k 0,r, ; w  k 0,r, T ; w
(A.4)

for all T  
and hence
(A.5)

 

k 0,r, ; w  k 0,r,   i; w  0 , i  1,2,K , n   .
We observe that in the case n  
(A.6)


k   1,r, t; w  0 , t    1,   2,K , n .
Thus the capital values of the process at the beginning of period   1 are negative when
it is operated for durations   1 or longer.
22
Step 3:
The trivial case   0 is excluded. We then see from (A.4) that

 

k 0,r, ; w  k 0,r,   i; w  0 , i  1,2,K ,  1 .
(A.7)
Therefore


k t,r, ; w  0 , t 1,K ,
(A.8)
and from (A.3) this result extends to


k t,r, ; w  0 , t  0, 1,K ,  .
(A.9)
Thus when the process is operated for its optimal duration, the capital values of the
process at the beginning of each period t  0, 1,K ,  are positive.
In particular,


k ,r, ; w 
(A.10)
q
0.
1 r
Step 4:
Differentiating (A.10) with respect to r, we see that


d  k ,r, ; w 
0.
dr
(A.11)
Step 5:
Using the notation in this Appendix, equation (2.4) in the text may be written as
(A.12)


k t,r, ; w 


qt  k t  1,r, ; w
1 r
or, equivalently,
23


k t  1,r, ; w 
(A.13)

.
qt 1  k t,r, ; w
1 r
Step 6:
Setting t   in (A.13) and differentiating with respect to r, we obtain
(A.14)






d  k   1,r, ; w 
k   1,r, ; w d  k ,r, ; w 


0
2
dr
dr
1 r
 
because of (A.9) and (A.11). Continuing in this manner, we see that
(A.15)


d  k t,r, ; w 
 0 , t  0, 1, K ,  .
dr
Step 7:
At some higher interest rate r  r , it is possible that
(A.16)


k T ,r  r, ; w  0 for some T   .
In this case at the new higher interest rate there is a new optimal duration    for
which
(A.17)


k t,r  r,  ; w  0 , t 0, 1,K ,  .
In view of (A.8) we see that
(A.18)


k t,r,  ; w  0 , t 0, 1,K ,    .
Hence the previous argument shows that all of these must fall when r rises to r  r .
Step 8:
We conclude that the present discounted value of the production process, which
Hicks denotes simply by k0 , is initially positive and falls continuously with increasing r,
even if the duration becomes shorter with increasing r. Thus there exists a unique r  r 0
and a corresponding optimal duration  0 such that the present discounted value of the
production process at the beginning of period 0 is equal to 0:
24


k 0,r 0 , 0 ; w  0 .
(A.19)
The interest rate r  r 0 is the unique internal rate of return for the production process at
the wage rate w.
Moreover, if (A.19) holds and r 0 is increased to r 1  r 0  r so that
(A.20)




k 0,r 1 , 1; w  0 ,
there exists some w1  w such that
(A.21)
k 0,r1 , 1; w1  0
where the optimal duration at interest rate r 1 is 1  0 . This establishes that the factorprice curve for the production process is downward sloping.
25
Appendix B
Here we sketch a generalization of the original von Neumann model [von
Neumann, 1938, 1945-46] to allow for labor as a primary factor of production.
There are m different activities for producing n different
commodities m  n or m  n . Activity j operated at the unit intensity level requires a




labor input a0 j and a vector of commodity inputs a1 j ,K , an j to produce a vector of


commodity outputs b1 j ,K ,bn j . We define the vector of labor requirements
(B.1)


 a11 L

A  M
a L
 n1
a1m 

M,
an m 
 b11 L

B M
b L
 n1
b1m 

M .
bn m 
A0  a01 , K , a0m ,
the input matrix
(B.2)
and the output matrix
(B.3)
The column vector
(B.4)
 x1 
 
x   M
x 
 m
give the intensity levels at which each of the m activities are operated. Finally, the vector
(B.5)
 C1 
 
C   M
C 
 n
represents the consumption of commodities.
26
Labor grows at the exogenous rate g  0 . The system is capable of growth at rate
g if
(B.6)
1  g Ax  Bx  C
with C  0, C  0 and
x  0, x  0 .
The row vector the n commodity prices is
(B.7)

p  p1 , K , pn

n
p
with the normalization
i
1,
i1
the wage rate is w  0 , and the steady-state rate of interest is r  0 . A steady-state
equilibrium at a given value of r is possible if there is a solution to the dual von Neumann
price system satisfying
(B.8)
 
wA0  1  r p A  p B with
w  0, p  0, p  0 .
If the cost of operating an activity exceeds its revenue, that activity is shut down (is
operated at a zero intensity level):
(B.9)
x j  0 if
  p a   p b
n
wa0 j  1 r
n
i
ij
i ij
i1
,
j 1,K ,m .
i1
If an activity is operated at a positive intensity level, then revenue must exactly cover
cost:
(B.10)
  p a   p b
n
wa0 j  1 r
n
i
ij
i1
if
i ij
x j  0,
j 1,K ,m .
i1
Similarly, the price of a commodity is zero if it is in excess supply:
(B.11)
1 g  a
m
pi  0 if
j 1
n
x   bi j x j  Ci ,
ij j
j 1,K ,m .
i1
And if a commodity has a positive price, its supply and demand are equal:
1 g  a
m
(B.12)
j 1
n
x   bi j x j  Ci
ij j
if
pi  0, i 1,K ,n .
i1
We denote employed labor by L  A0 x . Then by combining (B.9)—(B.12) we see
that in a steady-state equilibrium
27
1  g pA x  p C  p Bx  wL  (1  r) Ax .
(B.13)
Denoting the per capita value of capital by v 
consumption by pc 
(B.14)
p Ax
and the per capita value of
L
pC
, equation (B.13) may be rewritten as
L


pc  w  r  g v .
The latter may also be written in the more familiar form
(B.15)
Consumption + Net Investment  pc  gv  Wages + Profits  w  rv .
This is a generalization of the result derived by Hicks in his “Social Accounting” chapter
[Hicks, 1973, pp. 28-36].
28
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34
Footnotes
i
An earlier draft of this paper was presented to a joint meeting of the Economic Theory and the History of
Political Economy Workshops at Duke University, and I am grateful for the comments received there. My
special thanks goes to Neil De Marchi for the many improvements that he suggested and to Daniel A.
Graham for his help and encouragement.
ii
The list of topics [Burmeister, 1974, pp. 413-414] was:
(1) truncation of production process and the uniqueness of the internal rate of return;
(2) factor-price curves and the factor-price frontier;
(3) reswitching (both as usually defined and along a dynamic path);
(4) duality results;
(5) determination of relative factor shares;
(6) complications arising from joint production (including an example showing why the
nonsubstitution theorem is invalid when certain types of joint production exist);
(7) technological change and the relationship of Hicks’ old classification, his new classification, and
Harrod neutrality;
(8) a clarification of the famous “Ricardo on Machinery” dispute;
(9) dynamic paths and stability (or the Traverse) when problems of uncertainty are circumvented;
(10) two numerical examples illustrating full-employment transitions;
(11) a simple demonstration that the neo-Austrian method is a special case of the more general von
Neumann approach;
(12) an interpretation of a neo-Austrian example without joint production as a specialized LeontiefSraffa model;
(13) the problem of substitution and a brief explanation of i) stability results from other sectoral
models, and ii) the similarity of the strong assumptions which are required for convergence;
(14) a very short summary of the subjects which traditionally have been controversial in capital theory
(including a paradox revealed by the reswitching controversy); and
(15) a generalized von Neumann model with consumption and a primary factor (presented in the
Appendix).
iii
Hicks assumes that inputs and outputs are both valued at the beginning of the week [Hicks, 1973, p. 20].
The more conventional assumption used here is that labor is paid and revenue from output is received at the
end of the period. Accordingly, Hicks’ capital values are related to ours by ktHicks  kt 1  r  . The critical
observation is that the equilibrium conditions k0Hicks  0 and k0  0 both define the exactly same relationship
between the real wage rate and the interest rate.
1
The exactly analogous relation given Hicks’ timing assumptions is ktHicks  qtHicks  ktHicks
; see [Hicks,
1 R
1973, equation (2.1), p. 20].
iv
v
Similarly in Appendix A we introduce the following more precise notation for kt : Given a real wage rate,
the value of a production process at the beginning of period t, when that process is operated through to
T
period T, is given by k (t , r, T ; w )   q (1  r )  ( i 1t ) , t  T  n .
i
i t
vi
In earlier work [Hicks, 1965] he refers to the same relationship as the wage-interest curve.
35
Using the notation in Appendix A, define the implicit function G  w, r   k (0, r, ; w)  0 . Then
vii

  a  1  r 
i 1
i
dr
G / w

  i 0
 0 because (1) by assumption ai  0 and at least one labor input is
dw
G / r
 k (0, r, ; w) / r
positive, and (2) we have already shown in Appendix A that, for a given real wage rate, the capital value of
the process at the beginning of period 0 always falls with a rise in the interest rate.
Earlier Hicks gave a different definition for the “average period of production” and claimed that it
always increased with decreases in the rate of interest; see [Hicks, 1946, Appendix to Chapter XVII]. This
claim was shown to be false by Samuelson [Samuelson, 1947, p. 188]. Subsequently work by Hicks reveals
agreement about these issues [Hicks, 1965, Chapter XIII].
viii
ix
A Simple Profile has input ac and output 0 for weeks 0,1,
weeks m, m  1, , m  n  1 .
x
, m  1 and it has input au and output 1 for
As far as I know, Leland was allowed to keep his prize.
xi
Some caution is required because our subscripts for the von Neumann activities are one more than the
Hicks subscripts. This is because our Activity 1 is associated with the Hicks production period 0, etc.
The set of parameters generating this example in which Activity 3 is shut down when r  1 is
a1  a2  a3  1 , b2  3 , and w  0.5 . However, if a3 is reduced to 0.5 , then the production process is never
truncated because W 0  2 .
xii
xiii
My more ambitious goal for near aggregation is to compute the explicit functional forms of both
aggregate capital and the per capita aggregate production function for specific economies with
heterogeneous capital goods, but this turns out to be an extraordinarily complex computational task. With
each improvement in the Maple engine, I have been able to make a little more progress, but there is still a
long way to go.
xiv
The Equal Organic Composition of Capital condition for a no-joint production neoclassical technology
is that, across steady-state equilibria, the ratio of the wage bill to the value of capital be the same function
of the interest rate for every industry.
xv
Those interested in this issue should not miss reading [Samuleson, 1959] and [Hagemann, 1994].
xvi
These results, along with the associated duality relationships, are proved in [Burmeister, 1976]. Also see
[Samuelson, 1975] where similar results were first stated without proof.
xvii
Also see the discussion of static expectations on p. 56.
xviii
It is implicitly assumed that an alternative feasible path exists which differs from the original
competitive path ct , kt Tt 0 only by  ct and  ct 1 . For some technologies such an alternative path may not
always exist, and then more complex methods are required to establish bounds on the interest rate.
xix
I am indebted to my colleague Neil De Marchi for this observation.
36
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